Georgescu, George; Popescu, Andrei Closure operators and concept equations in non-commutative fuzzy logic. (English) Zbl 1065.03016 Tatra Mt. Math. Publ. 27, 67-90 (2003). Fuzzy sets for a noncommutative framework are studied in this paper, namely, functions from a set \(X\) to a generalized residuated lattice \(L\). The latter are defined similarly to residuated lattices, but their monoidal operation is no longer assumed to be commutative.In analogy to the commutative case, \(L\)-closure operators on \(L^X\) and \(L\)-closure systems are defined and their equivalence shown. Furthermore, the notion of a Galois connection between sets \(X\) and \(Y\) is introduced, and it is shown how it may be characterized on the base of left- and right-\(L\)-closure systems on \(L^X\) and \(L^Y\). Finally, \(L\)-Galois connections induced by some binary \(L\)-relation on \(X \times Y\) are considered, and also how this relation may be recovered if the Galois connection is known only partly. Reviewer: Thomas Vetterlein (Dortmund) Cited in 2 Documents MSC: 03B52 Fuzzy logic; logic of vagueness 03E72 Theory of fuzzy sets, etc. 06A15 Galois correspondences, closure operators (in relation to ordered sets) Keywords:noncommutative fuzzy closure operator; noncommutative conjunction; concept lattice; concept equation PDFBibTeX XMLCite \textit{G. Georgescu} and \textit{A. Popescu}, Tatra Mt. Math. Publ. 27, 67--90 (2003; Zbl 1065.03016)